Foundations of Physics

, Volume 40, Issue 9–10, pp 1419–1428 | Cite as

On the Complementarity of the Quadrature Observables

  • Pekka Lahti
  • Juha-Pekka Pellonpää


In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon transform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.

Complementarity Quadrature observables Unsharp quadrature observables Phase space observables Radon transform Tomography observable Statistical method of moments 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Turku Centre for Quantum Physics, Department of Physics and AstronomyUniversity of TurkuTurkuFinland

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